Next: 12.5.4 Efficiency of Parallel Up: Fast Vortex Algorithm Previous: 12.5.2 Fast Algorithms

## 12.5.3 Hypercube Implementation

The global nature of the approach has made its parallel implementation fairly straightforward (see [Fox:88a]). However, as we have already seen in Section 12.4, that character was drastically changed by the fast algorithm as it introduced a strong component of locality. Globality is still present since the influence of particle is felt throughout the domain, but more care and computational effort is given to its near field. The fast parallel algorithm has to reflect that dual nature, otherwise an efficient implementation will never be obtained. Moreover, the domain decomposition can no longer ignore the spatial distribution of the vortices. Nearby vortices are strongly coupled computationally, so it makes sense to assign them to the same processor. Binary bisection is used in the host to spatially decompose the domain. Then, only the vortices are sent to the processors where a binary tree is locally built on top of them. For example, Figure 12.17 shows the portion of the data structure assigned to processor 1 in a 4-processor environment.

In a fast algorithm context, sending a copy of local data structure to half the other processors does not necessarily result in a load balanced  implementation. The work associated with processor-to-processor interactions now depends on their respective location in physical space. A processor whose vortices are located at the center of the domain is involved in more costly interactions than a peripheral processor. To achieve the best possible load balancing, that central processor could send a copy of its data to more than half of the other processors and hence itself be responsible for a smaller fraction of the work associated with its vortices.

Before a decision is made on which one is going to visit and which to receive, we minimize the number of pairs of processors that need to exchange their data structure. Following the domain decomposition, the portion of the data structure that sits above the subtrees is not present anywhere in the hypercube. That gap is filled by broadcasting the description of the largest group of every processor. By limiting the broadcast  to one group per processor, a small amount of data is actually exchanged but, as seen on Figure 12.18, this step gives every processor a coarse description of its surroundings and helps it find its place in the universe.

Figure 12.17: Data Structure Assigned to Processor 1

Figure 12.18: Data Structure Known to Processor 1 After Broadcast

If the vortices of processor A are far enough from those of processor B, it is even possible to use that coarse description to compute the interaction of A and B without an additional exchange of information. The far field of every processor can be quickly disposed of. After thinking globally, one now has to act locally; if the vortices of A are adjacent to those of B, a more detailed description of their vorticity field is needed to compute their mutual influence.

This requires a transfer of information from either A to B or from B to A. In the latter case, most of the work involved in the A-B interaction takes place in processor A. Obviously, processor B should not always send its information away since it would then remain idle while the rest of the hypercube is working. Load-balancing concerns will dictate the flow of information.

Next: 12.5.4 Efficiency of Parallel Up: Fast Vortex Algorithm Previous: 12.5.2 Fast Algorithms

Guy Robinson
Wed Mar 1 10:19:35 EST 1995